Question: Determine how many solutions exist for the system of equations. ${4x-2y = -2}$ ${2x-y = -1}$
Convert both equations to slope-intercept form: ${4x-2y = -2}$ $4x{-4x} - 2y = -2{-4x}$ $-2y = -2-4x$ $y = 1+2x$ ${y = 2x+1}$ ${2x-y = -1}$ $2x{-2x} - y = -1{-2x}$ $-y = -1-2x$ $y = 1+2x$ ${y = 2x+1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+1}$ ${y = 2x+1}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${4x-2y = -2}$ is also a solution of ${2x-y = -1}$, there are infinitely many solutions.